NEET Gravitation — Set 16

Question 1

Three masses of $ 8 \, \text{kg} $ each form an equilateral triangle with side $ 6 \, \text{m} $. What is the net force on one mass? ($ G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

Force between two masses: $ F = G \frac{m^2}{r^2} = 6.67 \times 10^{-11} \frac{8 \times 8}{6^2} = 1.185 \times 10^{-10} \, \text{N} $. Two forces at 60°: $ F_R = \sqrt{F^2 + F^2 + 2 F^2 \cos 60^\circ} $. $ F_R = \sqrt{(1.185 \times 10^{-10})^2 (1 + 1 + 1)} = 1.185 \times 10^{-10} \sqrt{3} $. $ F_R \approx 2.05 \times 10^{-10} \, \text{N} $.

Question 2

A $ 600 \, \text{kg} $ satellite orbits Earth at $ 8 R_E $ from the center. What is its total energy? ($ M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 $)

Accepted Answer

$ E = -\frac{G M_E m}{2 r} $. $ r = 8 R_E = 5.12 \times 10^7 \, \text{m} $. $ E = -\frac{6.67 \times 10^{-11} \times 6 \times 10^{24} \times 600}{2 \times 5.12 \times 10^7} $. $ E = -\frac{2.401 \times 10^{17}}{1.024 \times 10^8} \approx -2.34 \times 10^9 \, \text{J} $.

Question 3

A satellite orbits at $ 3 R_E $ from Earth’s center. What is its speed? ($ g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} $)

Accepted Answer

$ v = \sqrt{\frac{g R_E^2}{r}} $. $ r = 3 R_E $, $ v = \sqrt{\frac{9.8 \times 6.4 \times 10^6}{3}} $. $ v = \sqrt{2.09 \times 10^7} \approx 4.57 \times 10^3 \, \text{m/s} $.

NEET Physics: Gravitation — Practice Set 16

Q1. Three masses of \( 8 \, \text{kg} \) each form an equilateral triangle with side \( 6 \, \text{m} \). What is the net force on one mass? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q2. A \( 600 \, \text{kg} \) satellite orbits Earth at \( 8 R_E \) from the center. What is its total energy? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q3. A satellite orbits at \( 3 R_E \) from Earth’s center. What is its speed? (\( g = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q4. What is the gravitational force on a \( 7 \, \text{kg} \) mass \( 6 \, \text{m} \) from the center of a spherical shell of mass \( 300 \, \text{kg} \) and radius \( 4 \, \text{m} \)? (\( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q5. A \( 700 \, \text{kg} \) satellite orbits Earth at \( 10 R_E \) from the center. What is its total energy? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q6. A \( 5 \, \text{kg} \) mass is moved from \( R_E \) to \( 2 R_E \) from Earth’s center. What is the work done? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q7. A body is launched from Earth at \( 12.5 \, \text{km/s} \). What is its speed at infinity? (Escape speed = \( 11.2 \, \text{km/s} \))

Q8. A \( 8 \, \text{kg} \) mass is moved from \( 8 R_E \) to \( 16 R_E \) from Earth’s center. What is the change in potential energy? (\( M_E = 6 \times 10^{24} \, \text{kg}, R_E = 6.4 \times 10^6 \, \text{m}, G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \))

Q9. At what height above Earth’s surface is \( g \) reduced to \( 6.12 \, \text{m/s}^2 \)? (\( g_0 = 9.8 \, \text{m/s}^2, R_E = 6.4 \times 10^6 \, \text{m} \))

Q10. A body is launched from Earth at \( 11.5 \, \text{km/s} \). What is its speed at infinity? (Escape speed = \( 11.2 \, \text{km/s} \))